In another paper [l] measures determined from base functions, that have finite right and left limits everywhere and are of generalized bounded variation in the restricted sense, are studied and used to define non absolutely convergent integrals of Denjoy type. In this paper base functions of bounded variation and the corresponding measures are studied as a background for that paper. The results supplement parts of [2].

We show that increases with λ, for 0< λ < 1, being the kth zero of the ultraspherical polynomial and f(λ) a suitable function of λ. As a consequence, some inequalities for and an estimate for can be obtained.

All rings are associative. A ring T is said to be radical over a subring R if for every t ∈ T, there exists a natural number n(t) such that tn(t) ∈ R.

In [1] Faith showed that if T is radical over R and T is primitive, then R is primitive. We might then ask if the same is true if prime is substituted for primitive.

We construct two non-isomorphic nuclear, stably finite, real rank zero ${{C}^{*}}$-algebras $E$ and ${{E}^{'}}$ for which there is an isomorphism of ordered groups $\Theta :{{\oplus }_{n\ge 0}}{{K}_{\bullet }}(E;\mathbb{Z}/n)\to {{\oplus }_{n\ge 0}}{{K}_{\bullet }}({E}';\mathbb{Z}/n)$ which is compatible with all the coefficient transformations. The ${{C}^{*}}$-algebras $E$ and ${{E}^{'}}$ are not isomorphic since there is no $\Theta$ as above which is also compatible with the Bockstein operations. By tensoring with Cuntz’s algebra ${{\mathcal{O}}_{\infty }}$ one obtains a pair of non-isomorphic, real rank zero, purely infinite ${{C}^{*}}$-algebras with similar properties.

Let X and Y be two independent normal variates each distributed with zero mean and a common variance. Then the quotient X/Y has the Cauchy distribution symmetrical about the origin. Of particular interest in recent years has been the converse problem and examples of non-normal distributions with a Cauchy distribution for the quotient have been illustrated by Mauldon [9], Laha [2; 3; 4] and Steck [10].

A Bernstein–Walsh type inequality for ${{C}^{\infty }}$ functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak– Siciak theorem: a ${{C}^{\infty }}$ function on ${{\mathbb{R}}^{n}}$ that is real analytic on every line is real analytic; (2) Zorn–Lelong theorem: if a double power series $F\left( x,y \right)$ converges on a set of lines of positive capacity then $F\left( x,y \right)$ is convergent; (3) Abhyankar–Moh–Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero.

Let A be a finite-dimensional algebra over an algebraically closed field. By module is meant a finitely generated right module. A module T^ is called a tilting module if and there exists an exact sequence 0 → A^ → T' → T" → 0 with T'. T" direct sums of summands of T. Let B = End T^·T^ is called separating (respectively, splitting) if every indecomposable A-module M (respectively, B-module N) is such that either Hom^(T,M) = 0 or (respectively, N ⊗ T = 0 or . We prove that A is hereditary provided the quiver of A has no oriented cycles and every separating tilting module is splitting.

By an ortholattice we mean a lattice with 0 and 1 and a complementation operation which is an involutorial antiautomorphism. The free modular ortholattice on two generators has 96 elements—cf. J. Kotas [8].

This work focuses on the ongoing research of lineability (the search for large linear structures within certain non-linear sets) in non-Archimedean frameworks. Among several other results, we show that there exist large linear structures inside each of the following sets: (i) functions with a fixed closed subset of continuity, (ii) all continuous functions that are not Darboux continuous (or vice versa), (iii) all functions whose Dieudonné integral does not behave as an antiderivative, and (iv) functions with finite range and having antiderivative.

Given a centrally symmetric convex body $B$ in ${{\mathbb{E}}^{d}}$, we denote by ${{\mathcal{M}}^{d}}\left( B \right)$ the Minkowski space (i.e., finite dimensional Banach space) with unit ball $B$. Let $K$ be an arbitrary convex body in ${{\mathcal{M}}^{d}}\left( B \right)$. The relationship between volume $V\left( K \right)$ and the Minkowskian thickness (= minimal width) ${{\Delta }_{B}}\left( K \right)$ of $K$ can naturally be given by the sharp geometric inequality $V\left( K \right)\ge \alpha \left( B \right)\cdot {{\Delta }_{B}}{{\left( K \right)}^{d}}$, where $\alpha \left( B \right)>0$. As a simple corollary of the Rogers-Shephard inequality we obtain that ${{\left( _{d}^{2d} \right)}^{-1}}\le \alpha \left( B \right)/V\left( B \right)\le {{2}^{-d}}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach–Mazur distance to the regular hexagon.

A Newton polygon is introduced for a ramified point of a Galois covering of curves over a finite field. It is shown to be determined by the sequence of higher ramification groups of the point. It gives a blowing up of the wildly ramified part which separates the branches of the curve. There is also a connection with local reciprocity.

We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if ${{\mathcal{L}}^{\infty }}\left( \mathbb{G} \right)$ contains non-zero compact operators on ${{\mathcal{L}}^{2}}\left( \mathbb{G} \right)$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups $\mathbb{G}$ where ${{\mathcal{L}}^{1}}\left( \mathbb{G} \right)$ has the Radon-Nikodym property.

W. Kurc [5] has proved that in the unit sphere of Orlicz space LΦ(μ) generated by an Orlicz function Φ satisfying the suitable Δ2-condition and equipped with the Luxemburg norm every extreme point is strongly extreme. In this paper it is proved in the case of a nonatomic measure μ that the unit sphere of the Orlicz space LΦ(μ) generated by an Orlicz function Φ which does not satisfy the suitable Δ2-condition and equipped with the Luxemburg norm has no strongly extreme point and no H-point.

Repeated application of the unitary reflections

to the point (1,0) and line x + iy = 1 yields 24 points and 24 lines. These are the vertices and edges of the regular complex polygon 4 {3} 4 whose group has the abstract definition R4 = I, RSR = SRS [l]. The purpose of this note is to introduce the notion of regular quaternion polygon and give an example analogous to the preceding one.

An oriented link is positive if it has a link diagram whose crossings are all positive. An oriented link is almost positive if it is not positive and has a link diagram with exactly one negative crossing. It is known that the Rasmussen invariant, 4-genus, and 3-genus of a positive knot are equal. In this paper, we prove that the Rasmussen invariant, 4-genus, and 3-genus of an almost positive knot are equal. Moreover, we determine the Rasmussen invariant of an almost positive knot in terms of its almost positive knot diagram. As corollaries, we prove that all almost positive knots are not homogeneous, and there is no almost positive knot of 4-genus one.

A recent issue of the Bulletin contained a paper by Robillard [13] in which results from the theory of confounded factorial designs were used to obtain some lower bounds for Ramsey numbers. We shall derive, by more elementary methods, bounds which are much better than Robillard's in every example which he considered.

In Meir and Keeler [3], the authors proved a fixed point theorem in a complete metric space (X, d) for a mapping f that satisfies the following condition of weakly uniformly strict contraction:

Given ∊ > 0, there exists δ > 0 such that

(A)

Let X be a metrizable topological space and f:X→X a continuous selfmapping such that for every x ∈ X the sequence of iterates {fn(x)} converges. It is proved that under these conditions the following two statements are equivalent:

1. There is a metrization of X relative to which f is contractive in the sense of Edelstein.

2. For any nonempty f-invariant compact subset Y of X the intersection of all iterates fn(Y) is a one-point set. The relation between this type of contractivity and the Banach contraction principle is also discussed.

We discuss the $q$-exponential functions ${{e}_{q}},\,{{E}_{q}}$ for $q$ on the unit circle, especially their continuity in $q$, and analogues of the limit relation ${{\lim }_{q\to 1}}\,{{e}_{q}}\,((1-q)z)\,=\,{{e}^{z}}.$

In this paper, a concept of nearness convergence is introduced which contains the proximal convergence of Leader as a special case. It is proved that uniform convergence and this nearness convergence are equivalent on totally bounded uniform nearness spaces. One of the features of this convergence is that it lies between uniform convergence and pointwise convergence, and implies uniform convergence on compacta. Some other weaker notions of nearness convergence which are sufficient to preserve nearness maps are also discussed.